Plotting "the most dangerous equation in the world": http://press.princeton.edu/chapters/s8863.pdf. Monte Carlo test how the standard deviation changes with respect to the sample size.

mc_de_moivre.R

split.into.intervals <- function(what, interval.count = NULL, interval.size = NULL) {

  if (sum(sapply(list(interval.count, interval.size), is.null)) != 1)
    stop("exactly one of 'interval.count', 'interval.size', must be NULL")
  
  if (!is.null(interval.count))
    n <- length(what) / interval.count
  else
    n <- interval.size

  split(what, ceiling(seq_along(what) / n))
}

N <- 10000

xs <- seq(2, N / 2, by = 2)

X <- rnorm(N, mean = 0, sd = 10)

sds <- sapply(xs, function(n) {
  sd(sapply(split.into.intervals(X, interval.size = n), mean))
})

take <- 200
plot(xs[1:take], sds[1:take], type = 'l', col = 'blue', xlab = 'sample size', ylab = 'standard deviation')
lines(xs[1:take], (sd(X) / sqrt(xs))[1:take], col = 'darkgoldenrod2', lwd = 2)